with the \((a)_n = a(a+1)(a+2)...(a+n-1), (a) _0 = 1 \)  is the so-called
rising factorial or Pochhammer's Symbol [1]}.
In our present case \(b\)  has a fix non-negative integer value, so none of the solutions have poles at \(b = -n\) .
For the Kummer function \(M\)  when the parameter \(a\)  has negative integer numerical values (\(a = -m\) ) the solution is reduced to
a polynomial of degree \(m\)  for the variable \(z\) . In other cases \(a \ne -m\)  we get a convergent infinite series for all values of \(a,b\)  and \(z\) .
There is a connecting formula between the two Kummer functions,
\(U\)  is defined from \(M\)  via
\(U(a,b,z) = \frac{\pi }{sin(\pi b)}\left( \frac{M[a,b,z]}{\Gamma [1+a-b]\Gamma [b]}- z^{1-b}\frac{M[1+a-b,2-b,z]}{\Gamma [a]\Gamma [2-b]} \right),\) 
